consistent partial matching of shape collections via

Volume xx (200y), Number z, pp. 1–13

Consistent Partial Matching of Shape Collectionsvia Sparse Modeling

L. Cosmo1, E. Rodolà2, A. Albarelli1, F. Mémoli3, D. Cremers2

1University of Venice, Italy 2TU Munich, Germany 3Ohio State University, U.S.

Figure 1: A partial multi-way correspondence obtained with our approach on a heterogeneous collection of shapes. Our methoddoes not require initial pairwise maps as input, as it actively seeks a reliable correspondence by operating directly over thespace of joint, cycle-consistent matches. Partially-similar as well as outlier shapes are automatically detected and accountedfor by adopting a sparse model for the joint correspondence. A subset of all matches is shown for visualization purposes.

AbstractRecent efforts in the area of joint object matching approach the problem by taking as input a set of pairwise maps,which are then jointly optimized across the whole collection so that certain accuracy and consistency criteria aresatisfied. One natural requirement is cycle-consistency – namely the fact that map composition should give thesame result regardless of the path taken in the shape collection. In this paper, we introduce a novel approach toobtain consistent matches without requiring initial pairwise solutions to be given as input. We do so by optimizinga joint measure of metric distortion directly over the space of cycle-consistent maps; in order to allow for partially-similar and extra-class shapes, we formulate the problem as a series of quadratic programs with sparsity-inducingconstraints, making our technique a natural candidate for analyzing collections with a large presence of outliers.The particular form of the problem allows us to leverage results and tools from the field of evolutionary gametheory. This enables a highly efficient optimization procedure which assures accurate and provably consistentsolutions in a matter of minutes in collections with hundreds of shapes.

Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling—Shape Analysis

1. Introduction

Finding matches among multiple objects is a research topicthat has received a good deal of attention in recent years.In its most common formulation, it translates to the prob-lem of determining point-to-point maps between all shapesin a collection, subject to the requirement that the ex-tracted correspondence be in some way “consistent”. To

this end, a natural and widely accepted criterion is cycle-consistency [ZKP10], namely that composition of mapsalong loops in the collection should approximate the iden-tity. So far, the problem has been approached by indepen-dently computing pairwise maps [LH05, KLF11] betweenthe objects in the collection; the set of maps is then given asinput to a global optimizer which updates them so as to im-

submitted to COMPUTER GRAPHICS Forum (12/2015).

2 L. Cosmo et al. / Consistent Partial Matching of Shape Collections

prove their quality and produce a final, consistent correspon-dence. Although most of these approaches work well pro-vided that the input maps are sufficiently accurate, they suf-fer in the presence of noise (incorrect matches) in the mapsthemselves, or outlier (extra-class) shapes in the collection.Further, due to the combinatorial difficulty of imposing theconsistency requirement, many of the existing schemes pro-vide no guarantee that cycle-consistency is satisfied exactly.

In this paper, we introduce a new method for the jointmatching of multiple deformable shapes in a collection. Un-like the common approach outlined above, we do not requireany pairwise correspondence to be given as input, and in-stead formulate the problem as an optimization directly overthe space of cycle-consistent (multi-way) matches.

1.1. Related work

Probably the earliest attempt to tackle multiple shape match-ing in a principled way is the synchronistic matching ap-proach of Schmidt et al. [STCB07]. Given a collection ofplanar shapes, the authors model the joint matching prob-lem as the search of a shortest path in their product space.Due to the resulting intractability, the problem is relaxed toa series of pairwise sub-problems, and the cycle-consistencycriterion introduced as a regularizer. The method allows toimprove initial pairwise solutions, but consistency is not sat-isfied exactly and the method operates under the assump-tion that all shapes in the collection are similar. Pachauri etal. [PKS13] took a similar perspective by formulating thematching problem using the language of combinatorial op-timization; due to the spectral relaxation they perform, themethod tends to be sensitive to noise and outliers. Recently,Yan et al. [YLL∗14] formulated the problem as one of si-multaneous multi-graph matching [SRS13], but similarlyto [STCB07,PKS13], cycle-consistency is relaxed and grad-ually infused in a pairwise matching process as a regularizer.

Zach et al. [ZKP10] were probably the first to makean explicit attempt at finding solutions meeting the cycle-consistency requirement. Starting from an initial graph ofpairwise associations among the objects in the collection,they detect and remove erroneous edges as the ones givingrise to inconsistent loops in the graph. As an extension tothis approach, Nguyen et al. [NBCW∗11] apply global op-timization to select cycle-consistent maps while at the sametime allowing edges to be replaced by better map composi-tions. The method performs well when the full point-to-pointcorrespondence is known and accurate for all pairs of ob-jects. Huang et al. [HZG∗12] improved upon [NBCW∗11]by allowing sparse correspondences, and later rephrased theproblem by replacing the pairwise maps with a spectralcounterpart [HWG14]; however, the approaches do not ap-ply when the shapes being matched are only partially sim-ilar [RCB∗16]. Finally, Sahillioglu and Yemez [SY14] pro-posed a greedy algorithm that seeks nearly-isometric con-sistent solutions across all shapes in the collection. The ap-

proach only works well when matching shape extremities,and it is susceptible to outlier shapes and partiality. In par-ticular, its accuracy depends on the specific ordering of theshapes in the collection.

All the methods outlined above demonstrate good prac-tical performance in controlled settings, however there hasbeen a general lack of theoretical guarantees that ensure cor-rectness of the final correspondence under unfavorable con-ditions. First steps in this direction are taken by Huang andGuibas [HG13], who formulate a convex relaxation to thejoint matching problem using the language of semi-definiteprogramming. The authors derive theoretical guarantees onthe recovery of the correct joint correspondence from noisyinput maps [KLF11]. Very recent works in the field of infor-mation theory explore this direction more abstractly [CG14],giving conditions for perfect recovery under large outlier ra-tios. Chen et al. [CGH14] and Kezurer et al. [KKBL15] re-cently applied this analysis to consistently match partiallysimilar objects from a small fraction of densely corruptedpairwise maps. To our knowledge, their algorithms currentlyrepresent the state of the art within this family of approaches.

1.2. Contribution

In this paper, we introduce a novel technique to construct ac-curate, consistent correspondences within shape collections.Our formulation has the following key properties:

• The method operates by optimizing directly over thespace of cycle-consistent correspondences, without re-quiring pairwise maps to be given as input. As a result,cycle-consistency is satisfied exactly by construction.

• We employ sparsity techniques in order to cope with par-tially similar as well as outlier shapes in the collection –an aspect that has received limited interest so far, but thatcan frequently occur in practical scenarios.

• Our proposed method is easy to implement. Further, itcompares favorably with the state of the art on challeng-ing datasets while being orders of magnitude faster.

2. Preliminaries

We model shapes as compact two-dimensional Riemannianmanifolds Si (possibly with boundary) embedded in R3,equipped with the intrinsic distance function di.

Let us be given a collection C = {S1,S2, . . . ,Sn} of nshapes. The product space S1×·· ·×Sn consists of all pos-sible n-way (i.e., joint) matches between the shapes in C.However, in practical settings it is often the case that out-liers (e.g., shapes belonging to different classes) as well aspartially similar shapes (e.g., man and centaur) are present inthe collection (see Fig. 2). In order to deal with such cases,we extend the set of possible joint matches as follows. For-mally, we consider the set constructed as the union

Γ =⋃k∈I

∏j∈k

S j , (1)


L. Cosmo et al. / Consistent Partial Matching of Shape Collections 3

A B C

Figure 2: A collection of shapes may carry partiality at dif-ferent levels. Our method allows to extract consistent corre-spondences reliably under partial similarity (e.g., S1/S2/S3)and at the same time detect and avoid outlier shapes (S5).

where I is a collection of index sets k defined by the powerset (denoted by P) relation

I = {k : k ∈ P({1,2, . . . ,n})∧|k|> 1} . (2)

In other words, Γ is the set of all possible m-fold Carte-sian products between the shapes in C, with 1 < m≤ n.Clearly, this set also includes S1×·· ·×Sn and in particu-lar |Γ| grows exponentially with the number of shapes. Eachelement γ ∈ Γ with |γ|= d ≤ n now represents a joint matchbetween a subset of d shapes from the collection.

Definition 1 We define a multi-way match among d ≤ nshapes to be any element γ ∈ Γ. A multi-way match is repre-sented as the ordered d-tuple

γ = (pi)i∈k with k ∈ I and pi ∈ Si ,

where k is a sequence of shape indices, denoting the shapesmatched by γ. We will write pi ∈ γ to say that the vertex piis matched via γ.

Note that two multi-way matches γ,γ′ ∈ Γ may in generalhave different lengths |γ| and |γ′|. In particular, they mayor may not have shapes in common. We will therefore de-fine the overlap γ∩ γ

′ as the longest common subsequenceof their shape indices. For example, in Fig. 2 we show themulti-way matches A,B,C ∈ Γ. For A and B we have theoverlap A∩B = (1,2), whereas A∩C = (3) and B∩C = ∅.

For our purposes, we are interested in subsets of Γ thatsatisfy certain properties, as described in the following:

Definition 2 A multi-way correspondence between the nshapes in C is a subset Γ⊂ Γ satisfying: for every Si ∈ Cand for every pi ∈ Si, there exists at least one γ ∈ Γ such thatpi ∈ γ.

The above definition ensures that in a multi-way correspon-dence each vertex of each shape is matched to correspond-ing vertices on (a subset of) the other shapes. We now definewhat is the meaning of cycle-consistency in our setting.

Definition 3 We say that a multi-way correspondence Γ

between shapes in the collection C = {Si}ni=1 is cycle-

consistent if, for any j,k, `∈ {1, . . . ,n}, whenever Γ matchesp j ∈ S j to pk ∈ Sk and matches pk to p` ∈ S`, then Γ alsomatches p j to p`.

Remark 1 A multi-way match is always cycle-consistent byconstruction, since it is an element of a product set. Thisapplies to any cycle, with length possibly longer than 3.

Note that while individual multi-way matches are alwayscycle-consistent, a fixed point pi ∈ Si might be mapped tomultiple points on the other shapes by a multi-way corre-spondence. We therefore introduce the following notion:

Definition 4 Two distinct multi-way matches γ,γ′ ∈ Γ aresaid to be incompatible whenever pi ∈ γ and pi ∈ γ

′ for somepi ∈ Si and i ∈ {1, . . . ,n}.

An illustration of incompatible matches is given in Fig. 3.

p1p2

p3

q1

Figure 3: Left: The red matches violate cycle-consistency,since p1 6= q1. Right: Example of two incompatible multi-way matches (red and green): the hand in the middle shapeis assigned to multiple distinct points on the other shapes.

3. Problem statement

The goal of joint object matching is to determine a (possiblydense) correspondence among multiple shapes in a collec-tion, with the requirement that the correspondence be consis-tent along cycles of any length. In this Section we formulatethe joint matching problem as one of minimum-distortioncorrespondence [Mém11]. Differently from most other ap-proaches [STCB07, ZKP10, NBCW∗11, HZG∗12, PKS13,SY14, YLL∗14] our formulation comes with the theoreticalguarantee of cycle-consistency, and additionally deals withpartially similar as well as outlier shapes in a natural way.

Metric distortion. Suppose we are given two multi-waymatches γ,γ′ ∈ Γ respectively putting |γ| and |γ′| points intocorrespondence, where in general |γ| 6= |γ′|. Then, we canquantify the quality of the correspondence by the cost func-tion ε : Γ×Γ→ R+∪{∞} defined as:

ε(γ,γ′) = maxpk ,p`∈γ

p′k ,p′`∈γ′

|dk(pk, p′k)−d`(p`, p′`)| . (3)

Here we tacitly assume that the multi-way matches are com-pared only on their overlap, i.e., over the shapes in common.In (3) we put ε(γ,γ′) =∞ whenever γ and γ

′ are incompat-ible (see Fig. 3) or non-overlapping. This definition of costencodes the maximum metric distortion attained by the twomulti-way matches across the shape collection.

Multi-way Lp distortion. A multi-way correspondence



Γ⊂ Γ can be alternatively modeled as a binary functiong : Γ→{0,1} such that for every Si ∈ C and for every q∈ Si,

∑γ∈Γ

s.t. q∈γ

g(γ)≥ 1 . (4)

Function g can be seen as an indicator function over thespace of all possible multi-way matches. Then, the conditionabove simply asks that each point in each shape is containedin at least one γ ∈ Γ for which g(γ) = 1, thus being a strictrequirement to match all points in all shapes.

The overall metric distortion caused by a correspondenceΓ can be measured by the Lp distortion:

‖ε‖pLp(g×g) = ∑

γ,γ′∈Γ

εp(γ,γ′)g(γ)g(γ′) , (5)

with p ≥ 1. Now, determining a correspondence of mini-mum distortion amounts to seeking a minimizer (not uniquein general) to:

ming:Γ→{0,1}

‖ε‖pLp(g×g) (6)

where g ranges over all correspondences Γ⊂ Γ.

Example. In the specific case where n = 2, a multi-waymatch γ = (p1, p2) reduces to a pair of points and the errorcriterion of Eq. (3) simplifies to the absolute metric distor-tion ε((p1, p2),(q1,q2)) = |d1(p1,q1)− d2(p2,q2)|. Then,by taking the limit for p→∞ the expression (6) yields theclassical Gromov-Hausdorff distance between metric spaces(S1,d1) and (S2,d2) [Mém11].

Dealing with partiality. The combinatorial complexity ofoptimizing over all possible multi-way correspondencesΓ⊂ Γ makes the problem intractable even for small collec-tions. Partial remedy to this issue is provided by relaxing thebinary map to take continuous values, i.e., g : Γ→ [0,1].

We further note that, although Eq. (1) enables us to betterdeal with partially similar shapes, the constraint defined inEq. (4) requires us to match all points in all shapes. How-ever, we would like outlier shapes to not partake to the finalcorrespondence. Furthermore, we want to allow individualshape points to be left unmatched if they do not find suitablematches throughout the collection.

We model this requirement by introducing a sparse modelfor the correspondence. To this end, we relax condition (4)by demanding ∑γ g(γ) = 1 over Γ. This requirement givesus an interpretation of g as a discrete probability distributionover the space of all multi-way matches. Importantly, the L1-like constraint on g has a sparsity-promoting effect on thesolution, hence modeling partiality.

Unfortunately, directly minimizing a problem of theform given in Eq. (6) subject to ∑γ g(γ) = 1 would yieldtrivial solutions. Specifically, we can characterize the

global opt.local opt.

global minimizers by: g(γ) = 1 for γ=γ? and g(γ) = 0 otherwise, where γ

? istaken to be any γ ∈ Γ. This amountsto concentrating the whole mass of ginto one single multi-way match, as il-lustrated in the inset figure.

We sidestep this issue by passing to a maximization prob-lem. Suppose we are given, as opposed to the cost ε, a simi-larity function s : Γ× Γ→R+ measuring the extent to whichtwo given multi-way matches preserve pairwise distances. Apossible choice is given by the Gaussian score:

s(γ,γ′) = e− 1

µ2 ε2(γ,γ′)

, (7)

where µ2 ∈ R+ is the variance of s. Note that s(γ,γ′) = 0whenever ε(γ,γ′) =∞; that is, incompatible matches are as-signed zero similarity. We get to the following optimizationproblem, which we consider throughout this paper:

Problem 1 (Partial multi-way correspondence). Given acollection of shapes C, we seek a partial multi-way corre-spondence among them as a maximizer to:

maxg:Γ→[0,1]

∑γ,γ′∈Γ

s(γ,γ′)g(γ)g(γ′) (8)

s.t. ∑γ∈Γ

g(γ) = 1 (9)

c(γ,γ′)g(γ)g(γ′) = 0 ∀ γ,γ′ ∈ Γ , (10)

where we set c(γ,γ′) = 1 if the two matches are incompati-ble, and c(γ,γ′) = 0 otherwise. Eq. (10) ensures that incom-patible matches will not appear in any local optimum.

The transition to a maximization problem has a regular-izing effect on its optima, as there are no trivial maximizersmeeting the constraints in this case.

Remark 2 Any local solution to Problem 1 satisfies the keyrequirements of a multi-way correspondence: 1) it is alwayscycle-consistent (by construction of Γ); 2) shape points areactivated at most once by the correspondence (by Eq. (10));and 3) partial matches are allowed (by Γ and Eq. (9)).

A note on symmetries. In case the shapes in the collec-tion carry bilateral symmetries, mapping either side wouldin principle yield the same optimum for Problem 1. In thispaper we deem correct such symmetric solutions as long asthey remain consistent across all pairs of shapes (see Fig. 4).

(a) (b)

Figure 4: (a) Incorrect correspondence due to inconsistenthandling of the symmetry. (b) Even if the solution is notorientation-preserving, symmetries are treated consistently.



Figure 5: Outlier shapes are automatically excluded by ourapproach, as they do not find support from the other shapesin the collection. Note that the human shapes appearing inthis example come from different datasets (TOSCA, SCAPE,SHREC’14). A subset of all matches is shown.

4. Optimization

Problem 1 is a non-convex quadratic program with O(|Γ|)variables; as such, it is in general very difficult to solve andto give guarantees on the optimality of the solution. In thisSection we develop an efficient strategy to get good localsolutions to this problem. The general strategy is to decom-pose it into two sub-problems: a robust process to get goodmatch candidates (Section 4.1), and a restriction of the orig-inal problem to the reduced feasible set (Section 4.2).

4.1. First sub-problem (reducing the feasible set)

The first sub-problem is aimed at reducing the size of the fea-sible set Γ to a smaller subset of “stable” candidates Λ⊂ Γ.Then, we will directly optimize Problem 1 over the reducedfeasible set Λ.

Outline. The general insight behind our formulation is that,given a collection of shapes, it is relatively easy and inexpen-sive to solve for one single multi-way match between them.Specifically, the idea is to seek for a multi-way match γ ∈ Γ

that maximizes a measure of point-wise similarity acrossseveral shapes, hence taking advantage of the stability in-duced by the whole shape collection. The final goal is tokeep in the feasible set only multi-way matches maximiz-ing this measure of similarity, since they are expected to beaccurate and stable against outliers, as shown in Fig. 5.

This problem can be formulated as a series of quadraticprograms with sparsity constraints (Eq. (11)), each yieldinga multi-way match γ ∈ Λ. Note that mapping constraints areimposed such that only cycle-consistent matches are allowedto be local optima.

Solving for a single multi-way match. Assume for simplic-ity that |Si| = N for all i = 1, . . . ,n. Further, let us be given

a point-wise similarity function τ : Sk× S` → R+, measur-ing the similarity of some descriptor defined at shape points(an example is given in Section 5.1). Note that this functionis not the same as the one defined in Eq. (7), which insteadmeasures the similarity between multi-way matches.

We introduce the vector x∈ [0,1]nN , representing a proba-bility distribution over all points in

⋃i Si. Then, consider the

L1-regularized non-convex quadratic program:

maxx≥0

x>Ax s.t. x>1 = 1 . (11)

Here, matrix A is a symmetric similarity matrix:

A =

0 S1,2 · · · S1,n

S1,2 0 · · · · · ·...

... 0 Sn−1,n

S1,n ... Sn−1,n 0

, (12)

where each symmetric block Sk,` ∈ RN×N contains the sim-ilarity values between the points in Sk and S`, accordingto function τ. The reason for the zero blocks along the di-agonal will become clear with Theorem 1. Note that thematrix above is not related to the block matrix appearingin [HG13, CGH14], which instead represents a collection ofpairwise maps (ideally permutations).

The key result of this Section is that the support of anylocal maximizer to the problem above (i.e., the set of pointsfor which xi 6= 0) is guaranteed to be a single partial multi-way match γ ∈ Γ between the shapes in the collection, as westate in the following theorem.

Theorem 1 Let x be a strict local maximizer of problem (11),where A = A> and Aii = 0 for all i = 1, . . . ,nN. Then,Ai j > 0 for all i, j such that xi 6= 0 , x j 6= 0.

Proof. See the Appendix.

According to Thm. 1, local solutions to (11) cannot simul-taneously activate points with zero similarity. This gives usa powerful means to restrict feasibility to solutions that acti-vate at most one point per shape: It is sufficient to set Ai j = 0whenever indices i and j correspond to points on the sameshape, i.e., matrix A must have zero blocks on the diagonal.

Remark 3 Since local solutions to problem (11) are guar-anteed to be multi-way matches, they are always cycle-consistent by definition.

A series of quadratic problems. Clearly, in order to con-struct the reduced set Λ ⊂ Γ we need a way to enumeratethe local optima of problem (11). We do so by solving a se-quence of problems of this form, each with a different datamatrix (12). Specifically, in each problem we compute simi-larities from a reference descriptor (or “query”) to all shapepoints, and we discard all dissimilar points.

Suppose we are given a collection Q of queries to com-pare against. A family of problems of the form (11) can



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⇒

WKS

= AQ

qk ∈ Q

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γ

γ′

γ′

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⇒

Figure 6: Our matching pipeline. First sub-problem (from left): Given a collection of shapes as input, a set Q of queries aregenerated (e.g., by farthest point sampling in the joint WKS space); we then compute distance maps (shown here as heat mapsover the shapes) in descriptor space from each shape point to each query qk ∈Q, and keep the vertices having distance smallerthan a threshold; finally, a single multi-way match is extracted by solving problem (11). Second sub-problem: The multi-waymatches extracted by iterating the previous step are compared using a measure of metric distortion; the final solution (in orange)is obtained by solving problem (13) over the reduced feasible set.

then be generated as follows. Given a query qk ∈Q, for eachshape Si ∈ C we only consider the vertices p ∈ Si such thatτ(p,qk) > ξ for some threshold ξ > 0. In other words, eachquery selects a different subset of vertices from each shape;since we can generate and solve as many problems (11) asthere are queries qk, we can proceed constructively and storeeach solution in our reduced feasible set Λ, which will havesize |Λ|= |Q|.

Note that each of these problems will be quite small, sincethe number N′ of shape points that are similar to each queryis significantly smaller than the total number of points N.We refer to Sec. 5.5 for an empirical evaluation. We alsonote that this approach is different from previous approacheswhich require pairwise maps as input or which require geo-metrically consistent samples to be pre-selected across theshapes [NBCW∗11, HZG∗12, HG13, SY14, CGH14].

Example. Suppose we are given a point descriptor functionf :

⋃i Si→Rm, providing an embedding of all shapes in Rm.

The query set Q can be defined implicitly by a k-means clus-tering or by farthest point sampling directly in Im f .

Numerical solution. It is worthwhile to note that problemsof this form have a natural interpretation from the point ofview of evolutionary game theory [ARBTP09, RBA∗12].We leverage this connection by adopting the infection-immunization dynamics algorithm [RBB11], an efficient lo-cal optimizer with convergence guarantees that exploits thespecific structure of problem (11).

Symmetries. In order to favor symmetry-consistentsolutions (Fig. 4), we assume tobe given left-right maps for theshapes in the collection, i.e., la-belings f : S→{L,R} associatingeach shape point to either side.The maps are then used to aug-ment the shape descriptors. Whilethere are robust approaches to perform this task [LKF12], forour purposes it is enough to have a rough estimate so as to

avoid obviously inconsistent solutions. We do so by lookingfor approximate rigid symmetries [PSG∗06] on a multidi-mensional scaling in R3 of a few farthest samples per shape.An example of such procedure is shown in the inset figure.Note that, compared to existing methods, this is a simplerrequirement than having input pairwise maps.

4.2. Second sub-problem (correspondence)

We are now ready to solve a smaller version of Problem 1 byreplacing Γ with the reduced set Λ. We proceed by directlyrewriting the problem in matrix notation.

Suppose we solved |Λ| = M instances (one per query) ofproblem (11), hence we have partial multi-way matches γifor i = 1, . . . ,M at our disposal. We can now compose thesimilarity terms s(γi,γ j) into a similarity matrix B ∈ RM×M

+such that Bi j = B ji = s(γi,γ j), and we set Bii = 0 for alli = 1, . . . ,M by Theorem 1. The correspondence function gcan simply be represented by a vector g ∈ [0,1]M . Similarlyto the previous case, we arrive at the quadratic program:

maxg≥0

g>Bg s.t. g>1 = 1 . (13)

Note that the mapping constraints (10), which impose thatincompatible matches cannot be part of the final solution, arealready incorporated in the data matrix B. This is because weset Bi j = 0 whenever γi and γ j are incompatible (by Eq. (7)).

A problem of this form for the simple case of two shapeswas previously considered in [RBA∗12]. Local solutionsto (13) (obtained again with [RBB11]) will be accurate, al-though sparse. However, since the candidate set Λ is likelyto contain good match hypotheses due to the previous opti-mization, there is hope to elicit a larger correspondence fromit. To this end, we consider three simple approaches:

Grouped sparse. Following [RBTP09, ART12], we pro-ceed by iteratively solving updated versions of problem (13).Whenever a local optimum is reached, the matches resulting



Data: Shape collection C of n shapesResult: Partial multi-way correspondence Γ among the

shapes in Cpre-processing (Sec. 5);Q← generated as in the Example of Sec. 4.1;Λ←∅;forall the q ∈ Q do

find points p ∈ Si s.t. τ(p,q)> ξ for i = 1, . . . ,n;construct similarity matrix A as in Eq. (12);x← solve problem (11) using [RBB11];γ← support of x;update Λ← Λ∪{γ};

endconstruct similarity matrix B as in Sec. 4.2;g← solve problem (13) “grouped” as in Sec. 4.2;Γ← support of g;

Algorithm 1: Full pipeline of our method for consistentpartial matching of shape collections. Detailed parametervalues are given in Section 5.

from the optimizer g are stored, and the data matrix is mod-ified by setting Bi? = B?i = 0 for all i such that gi 6= 0. ByTheorem 1, this amounts to reducing the feasible set to theremaining candidates in Λ. The iterations stop when the ob-jective g>Bg falls below a certain threshold.

Spectral relaxation. A different way to approach the prob-lem consists in replacing the L1 constraint g>1 = 1 by aL2 counterpart g>g = 1. This type of constraint acts as aTikhonov regularizer, which tends to yield denser solutionsfor this kind of problems. A global optimum can then becomputed by Rayleigh’s ratio as the principal eigenvector ofB. This comes at the price of sacrificing the mapping con-straints guaranteed by Theorem 1, which must be imposedby a post-processing of the obtained solution [LH05].

Elastic net. Finally, one may introduce a form of control-lable sparsity into the problem by elastic net regulariza-tion [RTH∗13]. In this case, the L1 constraint is replaced bythe convex combination (1−α)g>1+αg>g = 1, where pa-rameter α ∈ [0,1] allows to transition smoothly from a for-mulation equivalent to (13) (hence sparse) to a purely spec-tral solution (denser).

In Fig. 7 we show a full comparison of the three alter-natives on the TOSCA dataset, using the cumulative errormeasure defined in Sec. 5. Finally, the main steps describingour matching pipeline are given in Algorithm 1 and Fig. 6.

4.3. Complexity and scalability

We conclude the theoretical part with a complexity analysisof our method. Suppose our collection C is made of n shapes,each shape has N points, and M is the number of queries.

For a single query, computing the similarity matrix A

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TOSCA

Grouped(sparse( (169(matches)Spectral( (361(matches)Elastic(net(α=0.7((80(matches)

Elastic(net(α=0.9((176(matches)

Figure 7: Comparison between different numerical ap-proaches to solve problem (13). Left: Matches visualized ontwo scanned shapes from the SHREC’14 dataset, extractedfrom a multi-way correspondence of length 7; the spectralrelaxation yields 43 matches (in green), while the elasticnet with α = 0.7 only 11 matches (in red), although lessnoisy. Right: Quantitative comparison on the entire TOSCAdataset; the iterated L1 approach provides the best combi-nation of size and accuracy.

takes O((nN)2) operations. In practice, since for each querywe have N′ � N, this is a fast operation of the orderO((nN′)2). Optimization of problem (11) using evolution-ary dynamics [RBB11] is a O(nN′) step. The complexity ofthe first sub-problem (generation of Λ) is thus O(M ·(nN′)2).

Next, constructing matrix B is a O(M2) operation; thisalso involves computing geodesic distances among allpoints in Λ, which can be done efficiently via fast march-ing [WDB∗08]. Since optimizing problem (13) is a O(M)process, the overall complexity of the second sub-problem isO(M2). Note that in all our experiments we have once againM� nN, hence this step of the pipeline is typically very fast.We refer to Sec. 5.5 for an experimental evaluation.

5. Experimental results and applications

We performed a wide range of experiments on sev-eral benchmarks, namely: TOSCA [BBK08], SCAPE[ASK∗05], KIDS [RRBW∗14], and SHREC’14 [PSR∗14].These datasets consist of multiple classes of nearly-isometricshapes, with some intra-class variation in the case ofKIDS and SHREC’14. All datasets with the exception ofSHREC’14 come with ground-truth correspondences withineach category. In all the experiments, we ran our matchingalgorithm using M = 500 queries in descriptor space. Pa-rameter ξ was chosen as the 10th percentile of the descrip-tor distances to each query; the iterative process for solvingproblem (13) was stopped when the energy fell below 0.5. †

Pre-processing. WKS descriptors [ASC11] are precom-puted for all the meshes. We rescale each shape by the square

† Code will be made available at: http://vision.in.tum.de/members/rodola/code


http://vision.in.tum.de/members/rodola/code

http://vision.in.tum.de/members/rodola/code


0 0.05 0.1 0.15 0.2 0.250

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σ = 1.5e−5 (91)

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σ = 2.5e−5 (86)

σ = 3.0e−5 (76)

σ = 3.5e−5 (74)

σ = 4.0e−5 (74)

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µ = 5.0 (20)

µ = 6.0 (32)

µ = 7.0 (35)

µ = 8.0 (55)

µ = 9.0 (62)

µ = 10.0 (66)

µ = 11.0 (68)

µ = 12.0 (76)

Figure 8: Sensitivity experiments on a subset of TOSCA.Here we plot the error curves under different choices ofpoint-wise similarity parameter σ (left), and metric distor-tion parameter µ (right). In both graphs, the resulting num-ber of multi-way matches is reported in parentheses.

root of the k-th Laplacian eigenvalue, with k = 100 for allshapes; this has the effect of normalizing the meshes to havesimilar surface area (by Weyl’s law), and at the same time toyield comparable WKS. Where not specified otherwise, inour experiments we run the matching pipeline on N = 300farthest samples per shape (using the Euclidean metric). Thisis done in order to avoid solutions that unduly aggregate insmall regions. Note that we do not assume samples to becompatible across shapes as in [HG13], hence some localerror in the computed matches is to be expected.

Error measure. We quantify the quality of the correspon-dence by using the same measure of error defined in [HG13].Specifically, in our plots we show the percent of matches pewhich have geodesic error (i.e., distance from the ground-truth) smaller than a threshold e. This cumulative distri-bution is computed and aggregated over all the pairwisematches induced by the obtained multi-way correspon-dences. The geodesic error is normalized by the square rootof the area of each shape. As in [HG13], we also report val-ues for p0.16 and p0.02, which respectively capture the globaland local accuracy of the matching method.

5.1. Sensitivity analysis

The first set of experiments is aimed at analyzing the sensi-tivity of our method to different parametrizations. In orderto reduce overfitting, these experiments are performed on arepresentative subset of the TOSCA dataset, consisting ofthe victoria (12 shapes) and cat (11 shapes) classes.

Point-wise similarity. We measure the similarity betweenpoints on different shapes by the similarity between their as-sociated WKS descriptors. Each signature is computed onthe shape samples using 100 eigenpairs, 100 energy levelsand variance equal to 6.0 (default parameters as provided bythe authors). Given two points p ∈ Sk and q ∈ S`, we definetheir similarity by the Gaussian weight

τ(p,q) = e−1

σ2 ‖WKS(p)−WKS(q)‖22 . (14)

In Fig. 8 (left) we plot the error curves under different

Figure 9: Effect of parameter µ on the metric distortion termof Eq. (7). Increasing the value of µ makes geometric vali-dation more tolerant to distorted matches. In this real ex-ample on the SCAPE dataset, the colored regions show theadmitted metric distortion for a pair of multi-way matchesat different values of µ. An optimal value for this parametercan be chosen such that a prescribed metric distortion is notexceeded (e.g., constrained to the orange area).

choices of σ ∈ R. Note that smaller values of σ tend to yieldmore accurate solutions. The choice of σ also has an effect,although not very pronounced, on the final number of multi-way matches (reported in parentheses).

Metric distortion. As described in Eq. (7), penalizing themetric distortion of a pair of multi-way matches is done bymeans of a control parameter µ. As shown in Fig. 8 (right),changing the value of µ allows to control the size/accuracytrade-off of the final correspondence: as µ is increased, dis-torted matches are tolerated and included in the solution.Further illustration of this behavior is given in Fig. 9, wherewe show how the worst-case metric distortion over the shapecollection can be bounded by an appropriate choice of µ.The choice of this upper bound is ultimately driven by theapplication; e.g., it makes sense in shape exploration ap-plications (see Sec. 5.4) to require more accurate, althoughsparser matches in order to obtain a better clustering.

In a separate set of experiments, we investigate the effectof different similarity functions s. Namely, we consider boththe Gaussian function of Eq. (7) as well as a modified versionof it, given by replacing the worst-case cost of Eq. (3) by:

ε2(γ,γ′) = ∑

pk ,p`∈γ

p′k ,p′`∈γ′

|dk(pk, p′k)−d`(p`, p′`)|2 . (15)

Following [RBA∗12], we also include a relative (Lipschitz)notion of similarity in the comparison, defined as:

s(γ,γ′) =mink dk(pk, p′k)

µ

maxk dk(pk, p′k)µ . (16)

Since a fixed value of µ will in general scale differently inthe three cases, each variant is parametrized so as to yield30 multi-way matches on average. The results of this exper-iment are summarized in Table 1.



Figure 10: Left: Comparison between our method and thestate-of-the-art method of [CGH14] on the TOSCA andKIDS datasets; the quality of the input maps of MatchLift isalso reported. Right: Comparison of execution times. In allthe comparisons, the two methods generated a comparableamount of matches.

5.2. Comparisons

We compare our method with MatchLift, the convex relax-ation approach of Chen et al. [CGH14]. This method repre-sents, to the best of our knowledge, the state of the art forthis class of problems. In Fig. 10 we report the results on theTOSCA and KIDS datasets. Note that MatchLift did not pre-viously appear in these benchmarks. For a fair comparison,the input pairwise maps were computed using the methoddescribed in [CGH14] with WKS as a descriptor. In the samefigure we also report a runtime comparison of the two meth-ods on collections of increasing size. In Table 2 we showadditional comparisons with [HG13] and [HZG∗12] on theTOSCA and SCAPE datasets. As a baseline for standardpairwise matching, we also include the method of [RBA∗12]in the comparison. The results show that our method per-forms in line with the state of the art in most cases, with theadditional theoretical guarantee of cycle-consistency and ata fraction of the computational time.

We remind the reader that all methods included in thecomparisons, except for ours, require pairwise maps to begiven as input (also evaluated in the comparisons), hence act-ing more like global regularizers rather than “pure” multipleshape matching methods. Also note that our method was nottuned to perform well in the comparisons, as our sensitivityanalysis was only executed on a small subset of TOSCA.

L∞ L2 LipschitzLocal (p0.02) 26.81 16.01 20.04Global (p0.16) 96.21 95.49 91.05

Table 1: Comparison between different metric distortionmeasures on a subset of TOSCA. The best results (in bold)are obtained when we penalize the worst-case absolute met-ric error. Interestingly, there is no clear advantage in usinga relative error as opposed to its absolute counterpart.

Figure 11: Joint region matching on SCAPE (only a sub-set shown). The optimization process automatically excludedshape regions having incompatible segmentations with re-spect to the rest of the collection (e.g., two segments perarm). Our pipeline took 2 sec. to produce these results.

5.3. Region matching

Our method can be trivially modified to work with region-wise rather than point-wise correspondences, assuming a(possibly noisy) segmentation is provided for the inputshapes. The modification boils down to define a proper simi-larity measure among regions. To this end we use the simpleGaussian score of Eq. (14), where the cost term is replacedby the L2 distance between the area-weighted average WKSof each region. Regions are computed by consensus segmen-tation [RRBC14], using the code provided by the authors.

Note that since most shapes typically contain only 5 to 15regions, a full similarity matrix A can be constructed whichencodes the pairwise similarities among all regions in thecollection (i.e., we do not need to define queries). We canthen solve the resulting problem (11) iteratively, each timereducing the feasible set by removing solutions from the pastiterates (this is done by putting rows and columns of A tozero, as per Theorem 1). In Fig. 11 we show some qualitativeresults produced by this simple procedure when applied to anoisy version of SCAPE, in which 10 random outlier shapesfrom TOSCA were introduced.

Ours [HG13] [HG13]in [HZG∗12] [RBA∗12]

TOSCA (p0.16) 97.7 100 84.1 97.2 94.81SCAPE (p0.16) 95.9 99.1 83.2 99.3 91.10TOSCA (p0.02) 21.9 35.7 - 38.4 14.87SCAPE (p0.02) 50.6 42.1 - 44.4 10.29

Table 2: Comparisons with other recent methods in termsof global (p0.16) and local (p0.02) accuracy. The in columnreports the quality of the input maps [KLF11].



Figure 12: An example of shape retrieval. The query shapeis matched jointly to the shapes in the database, forming acluster with the shapes from the same class.

5.4. Other applications

Our approach is robust to the presence of outliers by design,and we can always extract an accurate solution as long as theoutliers do not have a structure.

Consider the example in Fig. 1. As problem (11) is iter-atively solved, the candidate set Λ is updated with matchesthat put the horse parts into correspondence, in addition tomatches that only relate the human bodies. The subsequentoptimization of (13) then extracts two intra-similar clustersof matches, one for each semantic group. In this case, it isclear that there is technically no reason to treat either of thetwo solutions as noise. Consider now a collection of shapesof a given class, which has been corrupted by introducingother shapes (Fig. 5). Since the extra-class objects fail toform stable matches with any other object in the collection,they will not appear in the final solution. This key feature ofour framework suggests, among others, two applications:

Number of shapes10 20 30 40 50 60 70

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Farthest point sampling100 200 300 400 500 600 700 800 900 1000

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Figure 13: Breakdown of our computational times over theSCAPE dataset. Left: Runtime as a function of a subset ofshapes in the collection, with 300 samples per shape. Thefirst and second optimizations refer to solving problems (11)and (13) respectively. The runtime for the first problem is ac-cumulated over M = 500 queries. Right: Runtime as a func-tion of farthest samples per shape, over the entire collection(71 shapes). Note the different scales among the two graphs.

Figure 14: An example of shape clustering of the TOSCAdataset, obtained by running the matching algorithm fol-lowed by extraction of connected components. Classes areencoded by color; note how all humans except for one vic-toria pose (in black) have been clustered together. Total run-ning time is around 1 min. 30 sec.

Shape exploration and clustering. Consider once again theexample in Fig. 5, and suppose both outlier shapes actuallybelong to the same class. This scenario can be seen as aninstance of structured noise – in fact, we now have two se-mantic classes forming intra-similar groups, and it would bedesirable to separate them into disjoint clusters [KLM∗12].We do so by a simple iterative procedure: 1) run Algorithm 1on the whole collection; 2) relabel the resulting multi-waycorrespondence into clusters, based solely on the shape in-dices; 3) remove the matched shapes from the collection andrepeat. Note that the clustering step is especially efficient, asit boils down to detecting connected components in a graphwhere each node represents a shape, and an edge exists be-tween two nodes whenever there exist (at least 3) matchesconnecting the respective shapes. Running this procedure onthe TOSCA dataset gives the results reported in Fig. 14.

Shape retrieval. The approach described above can be di-rectly applied to shape retrieval applications. Given a queryshape Sq /∈ C, the task is to detect the subset Cq ⊆ C con-taining shapes that belong to the same class as Sq. This canbe done by seeking a multi-way correspondence on the aug-mented set C ∪{Sq}, and by retaining the cluster of shapesthat match to Sq in the final solution (see Fig. 12).

5.5. Runtime

One of the key advantages of our matching method lies in itscomputational efficiency. In Fig. 13 we show a breakdownof the runtimes across the whole pipeline. Observe that thefirst optimization can be easily parallelized (we used 7 coresin our tests), as it amounts to solving independent instancesof problem (11), one problem per query.

Our method takes around 1 min. 30 sec. to match the en-tire SCAPE collection (71 shapes) when we use 300 sam-ples per shape. We note that, while from the point of view



of shape retrieval our method cannot compete with special-ized approaches, the increased running time accounts for thecorrespondences we obtain across all the shapes in the col-lection. In this regard, shape retrieval can be seen more as abyproduct of our method than an application per se.

Further runtime comparisons with the method of[CGH14] are given in Fig. 10. All experiments were codedin Matlab/C++ and run on an Intel Core i7 4900MQ with32GB memory, using publicly available code for [CGH14]and for the optimization step [RBB11].

6. Discussion and conclusions

In this paper we tackled the problem of consistent jointmatching of shape collections. Differently from the domi-nant approaches, we considered a situation in which the col-lection is not equipped with input maps between the shapepairs. To deal with this challenging scenario, we modeledthe problem as one of minimum distortion correspondenceacross the whole shape collection, while at the same time al-lowing outlier or partially similar shapes. We showed howto retrieve good local solutions to the resulting optimizationproblem by solving a sequence of quadratic programs in anefficient way – which in turn enabled favorable results in re-gion matching and shape exploration applications.

Our approach does have a few shortcomings. First, sinceour method relies on the computation of geodesics, we re-quire the shapes to have no significant missing parts, i.e.,shapes with large holes are not allowed. Second, while thesparse model allows to successfully deal with partial similar-ity at different levels, this partiality is not easily controllableand it might well be that incomplete matches are extractedeven within outlier-free collections. An example of this isshown in Fig. 14, where one human shape (in black) wasleft unmatched by our method. This is related to our neces-sity to establish a similarity criterion that acts globally onthe whole collection, hence driving longer, but less globally-similar matches to be cut out from the solution even if cor-rect. Enforcing specific shapes to partake in the final solutionis a possible direction of future work.

Acknowledgments

The authors wish to thank Mohamed Souiai, Zorah Läh-ner, Samuel Rota Bulò, Thomas Frerix, Luca Balsanelli, andValentina Gualtieri for valuable discussions. ER is supportedby an Alexander von Humboldt Fellowship.

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Appendix

The appendix contains a proof to Theorem 1. The proof fol-lows along the lines of, e.g., [ARBTP09, RBB11], althoughby taking a pure optimization perspective as opposed to agame-theoretical one.

Problem. We consider the constrained quadratic problem:

maxx≥0

x>Ax s.t. x>1 = 1 , (17)

where x ∈ [0,1]n, A = A> and Aii = 0 for all i = 1, . . . ,n.

First-order optimality conditions. A point x satisfies theKarush-Kuhn-Tucker (KKT) conditions for problem (17) if

there exist real constants λ and µ1, . . . ,µn with µi ≥ 0 for alli = 1, . . . ,n such that:

(Ax)i−λ+µi = 0 , (18)

and ∑i xiµi = 0. This latter condition further implies thatµi = 0 whenever xi 6= 0, since both quantities are nonneg-ative for all i = 1, . . . ,n. Thus, we can rewrite the KKT con-ditions (18) as

(Ax)i

{= λ if xi 6= 0≤ λ otherwise ,

(19)

for some λ > 0. We can easily see that

x>Ax = ∑i, j

xix jAi j = ∑i : xi 6=0

xi(Ax)i = ∑i : xi 6=0

xiλ = λ ,

(20)where the last equality follows from the constraint x>1 = 1.Hence, a point x satisfies the KKT conditions if

x>Ax≥ (Ax)i , (21)

for all i = 1, . . . ,n. This, in turn, implies x>Ax≥ y>Ax forall y satisfying y>1 = 1.

Second-order optimality conditions. A point x satisfies thesecond-order sufficiency conditions for strict local optimal-ity if x is a KKT point, and if the Hessian of (17) is nega-tive definite on the subspace M(x), that is z>Az < 0 for allz ∈M(x), where

M(x) = {z ∈ Rn : z>1 = 0 and z j = 0 for all j ∈ J}\{0} ,(22)

and

J = { j = 1, . . . ,n : x j = 0 ,µ j > 0} . (23)

This condition can be more compactly rephrased as follows.Consider the set

U = {y ∈ Rn : y>Ax = x>Ax ,y 6= x} , (24)

and let z = y−x with y∈U . Then (x+z)∈U , which meansz>Ax = 0, thus z ∈M(x); the converse is also true. Now wecan write

(x−y)>Ay =−z>A(x+ z) =−z>Az > 0 , (25)

by the negative definitess condition z>Az < 0. Thus, thesecond-order optimality condition can be succinctly phrasedas:

x>Ay > y>Ay (26)

whenever y>Ax = x>Ax and x 6= y. Note that the conversecan also be easily proven, i.e., Eq. (26) holds if and only if xis a KKT point and z>Az < 0 for all z ∈M(x).

We are now ready to prove the following theorem.

Theorem 1. Let x be a strict local maximizer of prob-lem (17), where A = A> and Aii = 0 for all i = 1, . . . ,n.Then, Ai j > 0 for all i, j such that xi 6= 0 , x j 6= 0.



Proof Assume Ai j ≤ 0 for i 6= j and xi 6= 0, x j 6= 0. Denoteby ei the i-th column of the identity matrix, and note that

ei>Ax = (Ax)i and ei>Ae j = Ai j. Now let y = δ(ei−e j)+x, where 0 < δ≤ x j. We have

y>Ax = δ(ei− e j)>Ax+x>Ax = x>Ax , (27)

where we used the fact that (ei−e j)>Ax= (Ax)i−(Ax) j =λ−λ = 0 by Eq. (19). However,

(x−y)>Ay = −δ(ei− e j)>A[x+δ(ei− e j)] (28)

= −δ2(ei− e j)>A(ei− e j) (29)

= −δ2(Aii +A j j−2Ai j) (30)

= 2δ2Ai j ≤ 0 , (31)

which implies x>Ay≤ y>Ay. In other words, we have con-structed a y for which the second-order condition (26) doesnot hold when y>Ax = x>Ax, contradicting the assumptionthat x is a strict local maximizer.


consistent partial matching of shape collections via

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